Disclosed is a damper for continuously variably adjusting a damping force that includes: a cylinder; a piston rod that is inserted into the cylinder and moves back and forth; a piston valve that is coupled to a bottom portion of the piston rod and moves back and forth in the cylinder according to the movement of the piston rod; a MR fluid that fills the cylinder and moves between a tension space over the piston valve and a compression space below the piston valve; and a volume compensation means that is located in the cylinder and compensates for a volume change of an inner space of the cylinder due to the movement of the piston rod, the volume compensation means including: a floating plunger that is configured to be below the piston valve and capable of moving up and down; a coil spring that is coupled to a bottom surface of the floating plunger and supports the floating plunger; and a support cup that is fixed onto the cylinder, coupled to a bottom portion of the coil spring to support the coil spring, and forms a buffer space, along with the floating plunger, between the support cup and the floating plunger.